36 ON THE MOTION OF SOLID BODIES THROUGH VISCOUS LIQUID [354
As an example of (32), we may suppose (as formerly for the plane) that F(i) = 0 from - oo to 0; V(t) = ht from 0 to r^\ V(t}=hrl} when t > TV Then ift<r1>
l 9vt 9i/4
and when t > rjy
(34)
When t is very great (34) reduces to its first term.
The more difficult problem of a sphere falling under the influence of gravity has been solved by Boggio (loc. cit.). In the case where the liquid and sphere are initially at rest, the solution is comparatively simple; but the analytical form of the functions is found to depend upon the ratio of densities of the sphere and liquid. This may be rather unexpected; but I am unable to follow Mr Basset in regarding it as an objection to the usual approximate equations of viscous motion.
§ 6. We will now endeavour to apply a similar method to Stokes' solution for a cylinder oscillating transversely in a viscous fluid. If the radius be a and the velocity V be expressed by V = Vn eint, Stokes finds for the force
7J, \^ ft • ••••'••t**«(l>»«*****\^ J
In (35) M' is the mass of the fluid displaced; k and k' are certain functions of m, where m = \ci \/(n/v), which are tabulated in his § 37. The cylinder is much less amenable to mathematical treatment than the sphere, and we shall limit ourselves to the case where, all being • initially at rest, the cylinder is started with unit velocity which is afterwards steadily maintained.
The velocity V of the cylinder, which is to be zero when t is negative and unity when t is positive, may be expressed by
simnt ,
------dn, ........................(36)
o n ' ^ '
in which the second term may be regarded as the real part of
1
n
dn ............................... (37)
^ '
We shall see further below, and may anticipate from Stokes' result relating to uniform motion of the cylinder, that the first term of (36) contributes nothing to F ; so that we may take
M' f
=-~-
Tr JQotion of the plane being periodic, he supposes that the plane and fluid are initially at rest, and that the plane is then (t = 0) moved with a constant velocity F.. The stream-function (ty) for this motion satisfies the same differential equation as does the transverse displacement (w'} of a plane elastic plate. And a surface on which the fiuid remains at rest (i/r = 0, d^/dn = 0) corresponds to a curve along which the elastic plate is clamped.